Complete graphs of odd order $K_{2n+1}$

Question

Are there any significant results in graph theory that are specific to complete graphs of odd order?

For example, I know one result for even order is: $K_{2n}$ has n-sun decomposition for all even $n > 2$.

Answer 1

A theorem of Häggkvist and Janssen proves the List Colouring Conjecture for all complete graphs of odd order. (That is, the "edge choosability" of the graph $K_{2n+1}$ is equal to its edge chromatic number.)

The conjecture may well be true for complete graphs of even order (and all other graphs) but that has not been proved yet, as far as I know.